Abstract How strongly can a positive precision update change a Gaussian response at a given information cost? Standard log-determinant and inverse-update identities do not by themselves determine the admissible costs when the worst modal contraction, total contraction, and active-rank cap are fixed. On a finite-dimensional physical space, for each directed Gaussian relative entropy separately, we derive its complete sharp attainable interval under these three constraints, including all equality and degenerate cases. A fixed probe also retains eigenvector orientation, but its normalized response is bounded by the largest contraction. Consequently, vanishing relative entropy forces every normalized individual response change to vanish, while absolute or extensive responses can still diverge if scale or rank is uncontrolled. We construct an exact finite hierarchical testbed for the same update problem on quotients of Z p d . A selected Taibleson-type hierarchy has closed radial graph and Green kernels; resistance recovers its cylinder tree, and baseline transfer resistance vanishes across separated cylinders. Point-shortcut response then reduces to endpoint valuations and an m × m matrix without allocating vertex-indexed objects. For the corresponding finite harmonic stiffness family with a common positive-definite mass matrix, we require no commutativity. An exact position-measurement bound holds at all temperatures, while a Bernoulli expansion through eighth order resolves the lowest-order inequivalent cyclic trace words within the expansion. For each fixed nontrivial quench, the joint-position data-processing gap is ( β ℏ ) 2 Tr [ ( Δ S ) 2 S 0 − 1 ] / 24 + O ( ( β ℏ ) 4 ) , whereas the full-state semiclassical correction D q − D cl relative to the classical limit begins at order ( β ℏ ) 4 ; asymptotic relative-entropy tightness in ratio follows as β ℏ → 0 .
Abstract How strongly can a positive precision update change a Gaussian response at a given information cost? Standard log-determinant and inverse-update identities do not by themselves determine the admissible costs when the worst modal contraction, total contraction, and active-rank cap are fixed. On a finite-dimensional physical space, for each directed Gaussian relative entropy separately, we derive its complete sharp attainable interval under these three constraints, including all equality and degenerate cases. A fixed probe also retains eigenvector orientation, but its normalized response is bounded by the largest contraction. Consequently, vanishing relative entropy forces every normalized individual response change to vanish, while absolute or extensive responses can still diverge if scale or rank is uncontrolled. We construct an exact finite hierarchical testbed for the same update problem on quotients of Z p d . A selected Taibleson-type hierarchy has closed radial graph and Green kernels; resistance recovers its cylinder tree, and baseline transfer resistance vanishes across separated cylinders. Point-shortcut response then reduces to endpoint valuations and an m × m matrix without allocating vertex-indexed objects. For the corresponding finite harmonic stiffness family with a common positive-definite mass matrix, we require no commutativity. An exact position-measurement bound holds at all temperatures, while a Bernoulli expansion through eighth order resolves the lowest-order inequivalent cyclic trace words within the expansion. For each fixed nontrivial quench, the joint-position data-processing gap is ( β ℏ ) 2 Tr [ ( Δ S ) 2 S 0 − 1 ] / 24 + O ( ( β ℏ ) 4 ) , whereas the full-state semiclassical correction D q − D cl relative to the classical limit begins at order ( β ℏ ) 4 ; asymptotic relative-entropy tightness in ratio follows as β ℏ → 0 .
A pure encoded state of an approximate quantum code is perturbed by exp(i eps W) with W drawn from a centered Gaussian ensemble, and D(eps) denotes the relative-entropy disturbance of a reduced state. This paper derives the complete fourth-order coefficient of the mean disturbance in closed form, and identifies what that coefficient is made of. The derivation has two stages, and they assume different things. The first assumes nothing about the noise beyond that it is centered and Gaussian: the coefficient splits into five channels, one disconnected drift term and four connected ones, each a finite sum over pairs of Karhunen-Loeve modes of the covariance contracted with exact confluent divided differences of the logarithm. For any concrete noise model the whole quartic disturbance is then computable deterministically, with no sampling. The second stage adds one hypothesis, that the mode family is complete and orthonormal, and evaluates every channel. The device throughout is to substitute the integral representation of the divided differences before performing the index sums, which turns each Kronecker delta into a coincidence of resolvent labels, and then to integrate by parts in the ring Q[d_1][t, m_1, ..., m_4] of resolvent traces. No coefficient is fitted and no index contraction survives. What survives is four scalar functions of the Schmidt spectrum: Xi, chi, S_w and T. The assembled coefficient is a function of d_1, d_2 and those four alone, and four is the right number. The five functions 1, Xi, chi, T, S_w are proved linearly independent for every d_1 >= 3, analytically: a degeneration lemma confines the kernel to two dimensions and an expansion about the degenerate spectrum closes it, with discriminant d_1 (d_1 - 2)^2 / 4, which vanishes exactly at d_1 = 2, where the two relations that do make the set reducible are exhibited. The four are not a list that happens to close. Each is a contraction of the Kubo-Mori metric and the Amari-Chentsov skewness tensor on two operator directions, the identity and the maximal-coherence operator of the Schmidt eigenbasis; and those directions are not chosen but read off the mode two-point function, which the Gaussian ensemble fixes identically in the spectrum and for every d_1. The two are Kubo-Mori orthogonal, so chi is the squared classical Fisher length of the spectrum and S_w - chi the squared coherent length, bounded below sharply by d_1^2 (d_1 - 1). Every channel is a rational-weighted sum of metric and skewness contractions, with d_2 the only number entering from outside the geometry, and the skewness tensor enters the whole coefficient exactly once: at order d_2^0 and with weight -1/6, which is precisely the weight the relative-entropy expansion assigns to its own cubic term. The paper is explicit about scope. A dedicated section separates what is proved about the ensemble alone, which is everything, from what the code supplies, which is the Schmidt spectrum and the reading of log d_2 as an area term, and from where the gap lies. On a family of Pauli strings satisfying the Knill-Laflamme conditions on a stabilizer code the first-order mode objects vanish identically, so three of the five channels are not small but exactly zero, and the entire Amari-Chentsov contribution goes with them: the mechanism is a property of the ensemble, not of approximate codes in general. The step from the coefficient to a physical reconstruction-error floor requires fixing the backreaction at a chosen fraction of its Page value, a modelling decision that this paper does not make. The limitations section states four further points, including one archived measurement at d_1 = 3, d_2 = 12 which is located at a definite configuration, quantified, and left unadjudicated with the two reasons given. Every closed form is certified. The checks run from the defining index sums, from the raw five-channel sum built from the mode objects on an explicit orthonormal basis, from the geometric identifications, and, beyond the invariant ensemble, against direct simulation on structured families and stabilizer codes. Two of the latter are retrodictions: coefficients measured and archived earlier in this programme, before the present theory existed, are reproduced to +0.4 and 0.0 standard errors, with a previously unexplained connected excess given exactly by the connected-channel formula. The appendix evaluates the eighteen connected contractions on which the connected channel rests, and the deposited code derives all eighteen of them from the Wick rule and then checks symbolically that they sum to the closed form. This paper supersedes and absorbs three earlier preprints of this programme: 10.5281/zenodo.23644208, 10.5281/zenodo.23684111 and 10.5281/zenodo.23726789, together with the contraction appendix of 10.5281/zenodo.23621356. Those records remain available and are not withdrawn; readers should cite the present paper for any of their results, which are proved here in full and without reliance on unrefereed material. Version 3 corrects the presentation of the appendix table of eighteen connected contractions. Up to version 2 the nine word pairs were listed in five grouped rows, on the ground that the relations T_3 = T_1^dagger and conjugation with u <-> v relate the members of each row; those relations preserve a pair's d_2-degree but not its value, and for three pairs the grouped entry is therefore not the pair's own. The table is now listed pair by pair, the collected sum before the kernel contraction is displayed, and the deposited script derives all eighteen entries from the Wick rule rather than taking them as given. No result of the paper changes: the two matchings of a word pair are added together, and sum_uv w_uv p_u = sum_uv w_uv p_v by the symmetry of w, so the collected sum and everything downstream of it are unaffected. Version 3 also corrects the wording of the proof of the entropy-kernel lemma. The first-order term of the expansion of Tr f(sigma) is Tr[Delta log sigma_0] + Tr Delta, and it is removed by the second, linear term of the relative entropy rather than by Tr Delta = 0 as the earlier versions said; the conclusion of the lemma is unchanged and is verified in the deposited code. A fifth script, an independent review pass over the statements the other four do not cover, is added to the archive. The verification scripts that generate every entry of the certification table, together with the derivation and symbolic verification of the appendix table, are included in the attached archive under the MIT licence.
We relate positive observable quotients to Page--Wootters (PW) clockdescriptions. Physicalization identifies a quotient by its kernel, andjointly faithful clock reductions recover the same exactness relation.Admissible PW data give a faithful functor into an independently definedquotient-clock category after representative lifts are identified modulophysical null directions. With algebraic lifts, it is full on its essentialimage at the level of retained state-and-clock data. A restricted converse characterizes strictly periodic, globallydeparametrizable realizations with unitary one-slice conditioning:one reference reduction must be unitary and the relative reductions mustform a strongly continuous periodic unitary group. The construction includesthe closed-operator domains required for unbounded generators. Retaining clock-event probabilities gives a separate strictness theorem.Normalizable stationary models with additive constraints and covariantperiodic clocks have state-independent clock marginals. A positivetwo-qubit constraint quotient violates this invariant despite invertiblereductions. This separates it from the additive class; its realization isitself interacting. We then define a broader stationary-conditioning class allowing generalizedcovariant clocks, unrestricted self-adjoint interactions, and separableenvironmental dilations. Its normal quantum-instrument data descendisometrically to positive quotients. Conversely, every normalized amplitudequotient-clock datum has an exact realization: a positive projectorconstraint selects the image of its isometric history map. Standardinstrument dilation extends the result to absolutely continuous normalcompletely positive instruments. For finite-dimensional physical and outputspaces with Sobolev-regular histories, a bounded finite-rank interactionsuffices by standard Hermitian completion and spectral isolation. Hencethis normalized operational sector has no obstruction to realization inthe broad class. For a fixed canonical periodic clock, we then bound the interactionremaining after arbitrary bounded clock-local Hermitian terms are allowed.Translation Fisher information gives a lower bound valid under separableenvironmental dilations. For instruments with orthogonal amplitude columns,a finite-rank construction attains the exact minimum while selecting theprescribed physical kernel. The modulated qubit family yields an explicitstrength--frequency threshold and a finite-event approximation-error bound. An exact Gaussian differential-image quotient coupled to a full oscillatorprovides an additive realization with matching physical metrics,conditioning, reconstruction, and relational probabilities, including finitereading intervals. These results distinguish restricted strictness frombroad representability; they neither identify all microscopic realizationsnor establish equivalence or strict superiority across every QRT and PWformulation.
We construct the covariance-sector Bures geometry of centered bosonic Gaussian states from the Gaussian symmetric-logarithmic-derivative equation and determine its quantum-admissible domain. At the Williamson uncertainty floor, the boundary geometry is anisotropic: spectrum-changing radial coefficients diverge, whereas squeezing and rotation directions along the pure Gaussian orbit remain finite. Dually, the covariance cometric acquires an exact m2-dimensional kernel when m modes become pure, yet the boundary remains at finite radial Bures distance. We derive the potential Φ=−21∑klog(νk2−41), which generates fixed-frame covariance dilation and equals βF on fixed-Hamiltonian thermal families. At large symplectic eigenvalue, the relative quantum correction is O(ℏ2/ν2), recovering covariance Fisher--Rao geometry. This intrinsic classical regime differs from the pure-state boundary, where classical statistics require a specified measurement channel. As a secondary comparison, the normalized Bures, Fisher--Rao, and action-matched Bures--Wasserstein determinant densities possess an exact one-mode junction at (x,η)=(φ,φ),x=2ν/ℏ. This junction is kinematic, and its branch interpretation is conditional. At a prescribed finite nonzero compression rate, the Bures action diverges while the Bures--Wasserstein cost remains finite; a bounded Bures budget instead enforces radial deceleration.
The fourth-order coefficient of the mean relative-entropy disturbance of a reduced encoded state under Gaussian coherent noise was assembled in closed form in previous work, but with two objects left as sums over spectral indices: the quartic commutator channel C, given there only as a four-index contraction of the third divided difference of the logarithm, and the diagonal part b_1 of the mixed channel. Both are closed here by one method, and the consequences are read off. The method. Substitute the integral representation of the divided differences before performing the index sums, which turns every Kronecker delta into a coincidence of resolvent labels, and then integrate by parts in the ring Q[d_1][t, m_1, ..., m_4] carrying the single derivation dm_k/dt = -k m_(k+1), where m_k(t) = sum_u (p_u + t)^(-k). Two structural facts make it usable, and ignoring either costs the solution. First, the integrand of chi = sum_u 1/p_u is exact, m_2 = -d(m_1)/dt, so chi is not an independent integrand but a boundary value; including it in the ansatz leaves its own coefficient undetermined. Second, no per-monomial convergence bound may be imposed on the antiderivative: only the total need converge, and the correct antiderivative contains four individually divergent monomials whose divergences cancel. Imposing the bound cuts the ansatz space from 111 monomials to 55 and the system then has no solution at all. The results are C = 1 - 2 Xi + (d_1^2/2 + 1/3) chi - (2/3) T - (d_1/6) S_w, b_1 = d_1^2 (Xi - chi), each with an explicit antiderivative, so no coefficient is fitted: the three coefficients of C emerge from the linear system as unknowns, and the additive constant and the chi term emerge from the two boundary values. No index contraction beyond the four functionals Xi, chi, T and S_w survives anywhere in the fourth-order coefficient. The functional S_w = sum_uv w_uv is not new to the theory: the connected channel already needs it at order d_2^2, in the form Conn-A_22 = 1 - Xi + D(Xi - 3)/4 + d_1 d_2^2 S_w/2. What is new is that the same functional returns in the d_2-independent part through C, that the two weights collect into (d_1/6)(3 d_2^2 - 1) S_w, that the coefficient needs exactly four spectral functionals and no fifth, and that S_w is not a linear combination of the other three. The last is proved for every d_1 >= 3, analytically: a degeneration lemma, sending one eigenvalue to zero and separating the coefficient space by the functional form of the log-coefficient, confines the kernel to two dimensions; an expansion about the degenerate spectrum then closes it, with discriminant d_1 (d_1 - 2)^2 / 4, which vanishes exactly at d_1 = 2. The same statement proves the irreducibility of the third functional T, which the preceding paper had established by least squares; and at d_1 = 2 the two relations S_w = chi + 2 Xi - 4 and S_w = -2T - 2chi are exhibited, which is why four functionals collapse to two there. The geometry. All four functionals are contractions of the Kubo-Mori (Bogoliubov-Kubo-Mori) metric g and of the Bogoliubov-Kubo-Mori skewness tensor tau, the Amari-Chentsov tensor of quantum information geometry, on two canonical directions: the identity 1 and the maximal-coherence operator J of the Schmidt eigenbasis, which is d_1 times the projector on the flat Schmidt superposition. Explicitly chi = g(1,1) = g(1,J), S_w = g(J,J), Xi = g(rho.J, J) and T = l_2[rho.J, J, J], with rho.J the Jordan product, and the numerical coefficients are contractions too: g(rho,rho) = 1 and g(rho,1) = d_1, the rank. The normalisation is fixed, not assumed, by the relative-entropy expansion S(rho + sX || rho) = s^2 g(X,X)/2 - s^3 tau(X,X,X)/6 + O(s^4). The directions are not a choice. In the two index patterns in which it enters, the mode two-point function of the invariant ensemble equals 2 (rho.J)_uv J_vu - 2 rho_uv rho_vu and 2 (rho.J)_uv 1_us - 2 rho_uv rho_vs identically in the spectrum, for every d_1; and a free index sum is a pairing with J since sum_uv X_uv = Tr[X J], while a delta-identified pair is a pairing with 1. Those two facts do not depend on the order of the expansion, so any contraction of the mode two-point function against any kernel l_r, at any order, lands in the span of {1, rho, J, rho.J}. In that language the identity and the pure coherence operator K = J - 1 are Kubo-Mori orthogonal, g(1,K) = 0, and S_w - chi = norm(K)^2: the fourth functional is the squared Kubo-Mori length of the coherence direction and chi is its classical Fisher part. Coherence enters through two functionals rather than one, since Xi - d_1 = g(rho.K, K) is purely off-diagonal as well. Every channel is then a rational-weighted sum of metric and skewness contractions on the four directions, with d_2 the only number entering from outside the geometry: G = g(Lambda,Lambda)/2, A_13 = -(5/6) D sum_a g(J_a,J_a), and b_1 = d_1^2 g(rho.J - 1, J). Sharper still, the Amari-Chentsov tensor enters the whole coefficient exactly once, at order d_2^0 and with weight -1/6, which is precisely the weight the relative-entropy expansion assigns to its own cubic term; everything else, at every order in d_2, is metric. Corollaries. The chain d_1^2 <= Xi <= d_1 sqrt(chi) <= chi holds for every spectrum, with equality at any one place exactly at a degenerate spectrum; the left link is from the preceding paper, the other two are new, and the consequence is that b_1 <= 0 always and vanishes only at degeneracy. The coherence length is bounded sharply, norm(K)^2 >= d_1^2 (d_1 - 1) and hence S_w >= d_1^3, saturated at the degenerate spectrum and nowhere else. The d_2-independent bracket collapses to (d_1^2/2 - 1/6) chi + T/3 - (d_1/6) S_w, in which Xi and the additive constant cancel exactly, so the constant term is independent of the functional that carries the whole second-order theory. At a degenerate spectrum C = 1 - (5/3) d_1^2 + (2/3) d_1^4, which gives 40 at d_1 = 3, and at d_1 = 2 the bracket equals 2 + (chi - Xi) and is therefore at least 2. Certification. Three independent routes to the full coefficient agree at every configuration tested: the raw five-channel sum evaluated from the mode objects on an explicit orthonormal Hermitian basis, the assembled equation of the preceding paper as printed, and the fully explicit form derived here. Three numbers published before these derivations existed are reproduced by them: C = 7.0492 at d_1 = 2 with p = (0.7, 0.3), C = 40.000 at d_1 = 3 with a degenerate spectrum, and B_1 = -10.2986 at d_1 = 2, d_2 = 2. The reduction of the four-index sum is an exact symbolic identity in the spectrum and in the integration variable; the antiderivative identity holds as a polynomial, not numerically; and the geometric form of C agrees with the four-index sum to 10^(-79). Two limitations are stated. Everything here is for the unitarily invariant ensemble, and nothing survives for a structured family such as a Pauli family satisfying the Knill-Laflamme conditions on a stabilizer code, where the first-order mode objects vanish and C = 0 identically. And one identity, an exact relation among the five functions holding on geometric spectra at d_1 = 3 and explaining why a least-squares test of reducibility is weak there, is established numerically with exact rational coefficients and held-out parameter values rather than symbolically; nothing else depends on it. One evaluative sentence of the preceding paper is corrected: the coefficient needs four spectral functionals, not three, and the fourth is printed in that paper's own assembled equation. The undercount is in the count. No equation, no theorem, no table entry and no value of that paper is affected, because C was computed there from its four-index sum directly and b_1 from its defining double sum directly. An expectation for order eps^6 is recorded and labelled as one: the directions are already forced for the part of the expansion built from first-order mode objects alone, while the two-point structures of the higher mode objects and the count of contraction patterns available to l_4 and l_5 remain open. Nothing at sixth order is computed here. This note is posted here for the first time. It closes the two residual index sums of Closed Forms for the Remaining Fourth-Order Channels of Gaussian Coherent Noise, and a Definitional Correction (Zenodo DOI 10.5281/zenodo.23684111), and corrects one evaluative sentence in it. The functional S_w it identifies geometrically was introduced in A Complete Fourth-Order Expansion of Reconstruction and Logical Errors for Arbitrary Gaussian Coherent Noise (Zenodo DOI 10.5281/zenodo.23644208), and the functionals Xi and chi in A chi-squared divergence law for the fourth-order breakdown of entanglement wedge reconstruction in Gaussian holographic codes (Zenodo DOI 10.5281/zenodo.23621356). The scripts reproducing every number in the note are deposited as metric_skewness_scripts.zip. They require Python 3.10+ with NumPy, SciPy, mpmath and SymPy; no other dependencies, CPU only. Five of them are reused unmodified from the deposit of the preceding paper, which is deliberate: the new closed forms are checked against code written before they existed.
We construct the covariance-sector Bures geometry of centered bosonic Gaussianstates directly from the Gaussian symmetric-logarithmic-derivative equationand determine its global quantum domain. The Robertson--Schr\"odingercondition bounds this domain below by the Williamson uncertainty floor. Itsboundary geometry is anisotropic: spectrum-changing radial coefficientsdiverge, whereas squeezing and rotation coefficients remain finite along thepure Gaussian orbit. Dually, the covariance cometric develops an exactm2-dimensional kernel when m Williamson modes are pure, although thefloor remains at finite radial Bures distance. We derive the radial potentialΦ=−21∑klog(νk2−41), which generates covariancedilation on fixed-Williamson-frame radial submanifolds and satisfiesΦ=βF on fixed-Hamiltonian thermal families. At large symplecticeigenvalue the relative symplectic correction is suppressed asO(ℏ2/ν2), yielding an intrinsic high-noise recovery of covarianceFisher--Rao geometry. This limit is distinct from the Williamson floor, whichis a quantum pure-state boundary rather than a classical limit. A classicalmodel at the floor instead requires an explicitly supplied measurement channelor commutative restriction; the boundary geometry constrains the structure onwhich that operational map acts but does not select the map itself. As asecondary comparison, we place the Gaussian Bures geometry alongside theFisher--Rao and Bures--Wasserstein covariance geometries. After introducing aconstant action-matching parameter η, their normalized determinantdensities possess an exact one-mode junction at(x,η)=(φ,φ). The equality is kinematic: it isneither an additional boundary nor a dynamical transition, and branch selectionrequires an explicitly stated minimum-log-determinant hypothesis. For the sameprescribed finite-rate radial compression toward the floor, the Bures kineticdensity and action diverge while the Bures--Wasserstein cost remains finite;under a bounded Bures kinetic budget, the intrinsic consequence is radialdeceleration rather than a forced change of geometry.
The complete fourth-order coefficient of the mean relative-entropy disturbance of a reduced encoded state under Gaussian coherent noise decomposes into five channels. Closed forms were previously available for two of them. This note evaluates the remaining three for the unitarily invariant ensemble, so that the whole coefficient is explicit for every bulk dimension d_1, every geometric dimension d_2 and every spectrum. The delta1-delta3 channel, reported in earlier work as measured but not evaluated, is A_13 = -(5/3) D (Xi - 1), with D = (d_1 d_2)^2 the total Hilbert-space dimension and Xi the Kubo-Mori spectral functional of the second-order theory. The proof uses only the two operator identities sum_a A_a^2 = D and sum_a A_a M A_a = Tr(M), and is independent of the code. The channel is negative for every spectrum, and not merely because Xi > 1: since the logarithmic mean is dominated by the arithmetic one, Xi >= d_1^2 with equality exactly at a degenerate spectrum, so A_13 <= -(5/3) D (d_1^2 - 1). The quartic commutator channel C is obtained in closed form as a contraction of the divided-difference kernel l_3 with two copies of the mode-sum tensor N, and is proved independent of d_2. The mixed channel B splits as B_1 + B_2. The remainder B_2 is the one that resisted: it is obtained here in closed form for every d_1, every d_2 and every spectrum, B_2 = 3 Xi - chi/2 - 2 + T, with chi = sum_u 1/p_u and T = sum_uvs l_2(u,v,s) p_u, where l_2 is the second confluent divided difference of the logarithm. T is a third spectral functional, and it is shown not to be a function of the other two: the best affine fit to {Xi, chi, 1} over forty interior spectra leaves worst residuals 2.86, 8.34 and 13.80 at d_1 = 3, 4, 5, against values of abs T reaching 82, 297 and 542. T is nevertheless not an alien object. In the variables m_k(t) = sum_u (p_u + t)^(-k) and h(t) = sum_u p_u/(p_u + t), the three functionals are Xi = integral h m_1 dt, chi = integral m_2 dt and T = -integral h m_1^2 dt, so T is the next member of one family. T and chi moreover scale alike, both as p^(-1), so no counting of degrees separates them; only the mode sum does. The derivation has two halves. The double mode sum Q_uvs = sum_ab (J_a)_uv (J_b)_vs (K_ab + K_ba)_su closes on the bulk spectrum alone, by four completeness identities: two that the preceding paper already prints, and two that carry an uncontracted mode operator and are what close the terms of K carrying A_a A_b Phi. Twenty-four cases are involved, and they are tabulated in the derivation note deposited in this record. The resulting divided-difference contractions then collapse by two identities proved from the integral representation of the Kubo-Mori kernels, F_4 = -d_1^2/2 and F_3 = d_1^2/2 - Xi. Three statements that could previously only be checked numerically follow as corollaries: B_2 is independent of d_2 and of the code; at a degenerate spectrum B_2 = -(1/2)(d_1^2 - 1)(d_1^2 - 4); and at d_1 = 2 it reduces to 2 (Xi - chi), that collapse being an accident of the dimension, since T there equals -(3/2) chi - w_12. Consequently the assembled coefficient is explicit at every configuration, its d_2-independent constant collapsing to 2 Xi - chi/2 - 1 + T + C. At d_1 = 2, d_2 = 2 and p = (0.7, 0.3) it reproduces a direct sampling measurement to 0.13 standard errors, at d_1 = 3 with a degenerate spectrum to 0.45, and the formula for A_13 reproduces retrodictively, to 0.55 standard errors, a measurement published before this derivation existed. Every closed form is certified against exact mode sums on an explicit orthonormal Hermitian basis of the full space, link by link rather than only end to end, and the formula for B_2 is tested on held-out spectra it never saw. An erratum to the preceding paper is also recorded. As printed, its third-order mode object L_abc = R([A_a,[A_b,[A_c,rho]]]) is anti-Hermitian, which would force A_13 to vanish identically for every ensemble, every code and every spectrum, contradicting the nonzero delta1-delta3 word measured in earlier work. The third-order deformation carries i^3, and restoring that factor removes the contradiction. That paper's own numerics evidently carried the factor, because it reports nonzero values of A_13 on two families that are not stabilizer codes; no value reported in any of the preceding papers is affected, and what requires correction is one definition as set. Two limitations are stated. Everything here is for the unitarily invariant ensemble, and none of the operator arguments survives for a structured mode family such as a Pauli basis on a stabilizer code. And a discrepancy with a published measurement at d_1 = 3, d_2 = 12 is located at a definite configuration but not adjudicated: the coefficient gives -9492 at the degenerate endpoint there, while the estimator used in this note cannot be carried to d_2 = 12 at the coupling used throughout, the Taylor control eps times the norm of W reaching 1.44. This note is posted here for the first time. It supplies the three missing channels of the five-channel decomposition of A Complete Fourth-Order Expansion of Reconstruction and Logical Errors for Arbitrary Gaussian Coherent Noise (Zenodo DOI 10.5281/zenodo.23644208), corrects one printed definition in it, and evaluates in closed form the delta1-delta3 word measured in A chi-squared divergence law for the fourth-order breakdown of entanglement wedge reconstruction in Gaussian holographic codes (Zenodo DOI 10.5281/zenodo.23621356). The degeneracy measurement it compares against is reported in Breakdown and Persistence of the Reconstruction-Error Hierarchy in Approximate Holographic Codes (DOI 10.13140/RG.2.2.19676.14723). The derivation note carrying the twenty-four case tables is deposited in this record as B2_derivation_note_2026-10-27.pdf. The scripts reproducing every number in the note are deposited as closed_forms_scripts.zip. They require Python 3.10+ and NumPy; no other dependencies, CPU only.